33. A = 1 4 8 -3 -7 -1 2 7 3 4 -2 29 5 5 369-5 -2 1 4 80 5 0250 −1 0 0 0 1 4 0000 0
33. A = 1 4 8 -3 -7 -1 2 7 3 4 -2 29 5 5 369-5 -2 1 4 80 5 0250 −1 0 0 0 1 4 0000 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Exercises 31–34 display a matrix A and an echelon form of A . Find a basis for Col Aand a basis for Nul A .
![In this example, we have a problem from linear algebra pertaining to matrices.
The matrix \( A \) is given as:
\[
A = \begin{bmatrix}
1 & 4 & 8 & -3 & -7 \\
-1 & 2 & 7 & 3 & 4 \\
-2 & 2 & 9 & 5 & 5 \\
3 & 6 & 9 & -5 & -2
\end{bmatrix}
\]
The matrix \( A \) undergoes row operations to reach its row echelon form:
\[
\sim \begin{bmatrix}
1 & 4 & 8 & 0 & 5 \\
0 & 2 & 5 & 0 & -1 \\
0 & 0 & 1 & 0 & 4 \\
0 & 0 & 0 & 0 & 0
\end{bmatrix}
\]
Explanation of the procedure:
1. The matrix \( A \) is a \( 4 \times 5 \) matrix.
2. Through a series of row operations (which typically includes row swapping, row multiplication, and row addition/subtraction), the initial matrix is transformed to a row echelon form.
3. The leading entries (also known as pivots), which are the first non-zero numbers in each row, are aligned in a stair-step fashion from left to right.
Importantly, the matrix can have zero rows at the bottom, indicating rows of all zeroes which are a result of the row operations to move to echelon form.
These steps help in solving systems of linear equations and understanding the properties of matrix transformations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F324f16c4-c4e5-4b06-b074-f261b39e021a%2Fefe72a24-33e3-4620-b0a6-e4dafafb6abe%2Fbh1njv_processed.png&w=3840&q=75)
Transcribed Image Text:In this example, we have a problem from linear algebra pertaining to matrices.
The matrix \( A \) is given as:
\[
A = \begin{bmatrix}
1 & 4 & 8 & -3 & -7 \\
-1 & 2 & 7 & 3 & 4 \\
-2 & 2 & 9 & 5 & 5 \\
3 & 6 & 9 & -5 & -2
\end{bmatrix}
\]
The matrix \( A \) undergoes row operations to reach its row echelon form:
\[
\sim \begin{bmatrix}
1 & 4 & 8 & 0 & 5 \\
0 & 2 & 5 & 0 & -1 \\
0 & 0 & 1 & 0 & 4 \\
0 & 0 & 0 & 0 & 0
\end{bmatrix}
\]
Explanation of the procedure:
1. The matrix \( A \) is a \( 4 \times 5 \) matrix.
2. Through a series of row operations (which typically includes row swapping, row multiplication, and row addition/subtraction), the initial matrix is transformed to a row echelon form.
3. The leading entries (also known as pivots), which are the first non-zero numbers in each row, are aligned in a stair-step fashion from left to right.
Importantly, the matrix can have zero rows at the bottom, indicating rows of all zeroes which are a result of the row operations to move to echelon form.
These steps help in solving systems of linear equations and understanding the properties of matrix transformations.
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